Depending on what kind of zeta functions you want, the [Selberg zeta function][1] allows you to relate lengths of closed geodesics to eigenvalues of the Laplacian.  In particular, you can use the Selberg zeta function in combination with a trace formula to prove the [prime geodesic][2] theorem for compact Riemann surfaces and get [Weyl's law][3].  This also leads to construct isospectral manifolds.

Similarly, for graphs one can look at the analogous [Ihara zeta function][4] to relate lengths of "geodesics" to certain spectral quantities.  In particular, one can get a characterization of [Ramanujan graphs][5] in terms of the Ihara zeta function.  There are also numerous variants to count different things in graphs (Bartholdi zeta function, path zeta functions), and I have a conjecture with Christina Durfee that zeta functions are better at distinguishing graphs spectrally than the usual (adjacency matrix or Laplacian) spectra considered.


  [1]: https://en.wikipedia.org/wiki/Selberg_zeta_function
  [2]: https://en.wikipedia.org/wiki/Prime_geodesic
  [3]: https://en.wikipedia.org/wiki/Weyl_law
  [4]: https://en.wikipedia.org/wiki/Ihara_zeta_function
  [5]: https://en.wikipedia.org/wiki/Ramanujan_graph